Pattern-avoiding permutation powers

Abstract

Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation π strongly avoids a pattern τ if π and π2 both avoid τ. They conjectured that for every positive integer k, there is a permutation in Sk3 that strongly avoids 123⋯(k+1). We use the Robinson–Schensted–Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least kk3∕2+O(k3∕logk) and at most k2k3+O(k3∕logk). We enumerate 231-avoiding permutations of order 3, and we give two further enumerative results concerning strong pattern avoidance. We also consider permutations whose powers all avoid a pattern τ. Finally, we study subgroups of symmetric groups whose elements all avoid certain patterns. This leads to several new open problems connecting the group structures of symmetric groups with pattern avoidance.